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Calculus

Higher-order Derivatives

Concept Quizzes

Second Order Derivatives
Characteristics of f, f', f''
Graphical Interpretation of Higher Order Derivatives
Third and Higher Order Derivatives
Higher Order Derivatives Problem Solving

Challenge Quizzes

Higher-order Derivatives: Level 1 Challenges
Higher-order Derivatives: Level 2 Challenges
Higher-order Derivatives: Level 4 Challenges

Characteristics of f, f', f''

Let $f$ be a function such that $f(0)=50.$ If the above graph represents the graph of the first derivative of $f$ on the interval $0<x<3,$ which of the following is the correct description of $f$ on this interval?

f

has exactly one inflection point at

x=0.

f

has a local minimum at

x=2.

f

is a monotonously increasing function.

f

is a monotonously decreasing function.

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by Brilliant Staff

For the function $y=x^3+x^2-x-1,$ determine the interval where $y$ is decreasing.

-\frac{1}{3} < x < -1

\frac{1}{3} < x < 1

-3 < x < -1

-1 < x < \frac{1}{3}

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by Brilliant Staff

Suppose $f$ is a function such that $f(0)=33$ and the first derivative of $f$ on the interval $-2 < x < 2$ is as shown in the above diagram. Which of the following is the correct description of the function $f$ on this interval?

f

is a monotonously decreasing function.

f

is a monotonously increasing function.

f

has an inflection point at

x=0.

f

has a local maximum at

x=0.

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by Brilliant Staff

Suppose $f$ is a function defined on the closed interval $-3 \le x \le 4$ with $f(0)=42,$ such that the graph of $f',$ the derivative of $f,$ on the interval is as shown in the above diagram. On what intervals is $f$ increasing?

-3 \le x < -2

-3 \le x \le 2

-2 \le x < 4

0 \le x < 4

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by Brilliant Staff

Suppose $f$ is a function defined on the closed interval $-3 \le x \le 4$ with $f(0)=3$ such that the graph of $f',$ the derivative of $f,$ on the interval is as shown in the above diagram. Find the $x$ -coordinates of the points of inflection of $f.$

x=-3, x=2

x=-3, x=4

x=-2, x=2

x=0, x=2

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by Brilliant Staff